Modified pulse sequence to estimate properties

ABSTRACT

Methods and related systems are described for estimating fluid or rock properties from NMR measurements. A modified pulse sequence is provided that can directly provide moments of relaxation-time or diffusion distributions. This pulse sequence can be adapted to the desired moment of relaxation-time or diffusion coefficient. The data from this pulse sequence provides direct estimates of fluid properties such as average chain length and viscosity of a hydrocarbon. In comparison to the uniformly-spaced pulse sequence, these pulse sequences are faster and have a lower error bar in computing the fluid properties.

BACKGROUND

1. Field

This patent specification relates to estimating properties fromexponentially decaying data. More particularly, this patentspecification relates to methods and systems for estimating materialproperties from a non-uniform sequence of excitations, such as with NMRmeasurements.

2. Background

Traditionally, fluid properties using Nuclear Magnetic Resonance (NMR)are obtained by study of longitudinal or transverse relaxation ordiffusion measurements. Longitudinal (T₁) relaxation data are acquiredusing an inversion-recovery pulse sequence where the recovery times areoften chosen arbitrarily. Transverse (T₂) relaxation data are acquiredusing the CPMG pulse sequence with pulses that are typically uniformlyspaced in the time-domain. There is no clear mathematical methodologyfor choosing the acquisition times at which the diffusion measurementsare obtained. The relaxation or diffusion data thus obtained from thesepulse sequences are used to infer the relaxation time or diffusiondistributions. In turn, these distributions are used to inferpetrophysical or hydrocarbon properties.

Traditional well-logging using NMR is employed to infer petro-physicaland fluid properties of the formation or in-situ hydrocarbon. Themulti-exponential time-decay of NMR magnetization is characterized byrelaxation time constants T₁, T₂ or diffusion D and correspondingamplitudes f_(T) ₁ (T₁), f_(T) ₂ (T₂) or f_(D)(D). Since the relaxationtimes and/or diffusion constants are expected to be continuous andtypically span several decades, these amplitudes are often referred toas relaxation-time or diffusion distributions. These distributionsprovide a wealth of information about both the rock as well as fluidproperties. See e.g. D. Allen, S. Crary, and R. Freedman, How to useborehole Nuclear Magnetic Resonance, Oilfield Review, pages 34-57, 1997.

An NMR measurement consists of a sequence of transverse magnetic pulsestransmitted by an antenna. Typically, the CPMG pulse sequence is used tomeasure T₂ relaxation. It consists of a pulse that tips hydrogen protons90° and is followed by several thousand pulses that refocus the protonsby tipping them 180°. The data are acquired by an antenna between theevenly spaced 180° pulses and are thus uniformly spaced in time. On theother hand, the T₁ relaxation is studied using the inversion-recoverypulse sequence. There is no specific rule for choosing the recoverytimes. See, Y. Q. Song, L. Venkataramanan, and L. Burcaw, Determiningthe resolution of Laplace inversion spectrum, Journal of ChemicalPhysics, page 104104, 2005. Similarly, there is no clear mathematicalframework to optimally choose the acquisition times for diffusionmeasurements.

The data acquired from all of the above pulse sequences can berepresented by a Fredholm integral (see, M. D. Hurlimann and L.Venkataramanan, Quantitative measurement of two dimensional distributionfunctions of diffusion and relaxation in grossly homogeneous fields,Journal Magnetic Resonance, 157:31-42, July 2002 and L. Venkataramanan,Y. Q. Song, and M. D. Hurlimann, Solving Fredholm integrals of the firstkind with tensor product structure in 2 and 2.5 dimensions, IEEETransactions on Signal Processing, 50:1017-1026, 2002, hereinafter“Venkataramanan et al. 2002”),

$\begin{matrix}{{M(t)} = {\int_{0}^{\infty}{{\mathbb{e}}^{{- t}/x}{f_{x}(x)}{\mathbb{d}x}}}} & (1)\end{matrix}$

where the variable x typically refers to T₁, T₂, or 1/D. See,Venkataramanan et al. 2002. Thus the measured data in eqn. (1) isrelated to a Laplace transform of the unknown underlying functionƒ_(x)(x).

Scaling laws established on mixtures of alkanes provide a relationbetween fluid composition and relaxation time and/or diffusioncoefficients of the components of the mixture (see D. E. Freed, Scalinglaws for diffusion coefficients in mixtures of alkanes, Physical ReviewLetters, 94:067602, 2005, and D. E. Freed, Dependence on chain length ofNMR relaxation times in mixtures of alkanes, Journal of ChemicalPhysics, 126:174502, 2007). A typical crude oil has a number ofcomponents, such as methane, ethane etc. Let N_(i) refer to the numberof carbon atoms or chain length of the i-th component. Let N denote themolar average chain length of the crude oil, also defined as theharmonic mean of the chain lengths,

$\begin{matrix}{{\frac{1}{\overset{\_}{N}} \equiv {\sum\limits_{i = 1}^{\infty}\frac{f_{N}\left( N_{i} \right)}{N_{i}}}} = \left\langle N^{- 1} \right\rangle} & (2)\end{matrix}$where f_(N)(N_(i)) denotes the mass fraction of component i in thehydrocarbon. According to the scaling law, the bulk relaxation timeT_(2,i) of component i in a crude oil follows a power law and dependsinversely on its chain length or number of carbon atoms, N_(i). Thus,the larger the molecule, the smaller the relaxation time T_(2,i). Therelaxation time also depends inversely on the average chain length N ofthe hydrocarbon, according toT_(2,i)=BN_(i) ^(−κ) N ^(−γ)  (3)

In eqn. (3), parameters B, γ and κ are constants at a given pressure andtemperature. Similarly, if D_(i) refers to the diffusion coefficient ofcomponent i,D_(i)=AN_(i) ^(−ν) N ^(−β)  (4)where A, ν and β and are constants at a given pressure and temperature.The scaling theory can also be used to derive an expression for fluidviscosity,η∝

D⁻¹

η∝ N ^(β)

N^(ν)

.  (5)

Traditional analysis of the measured NMR data to obtain fluid propertiesgoes as follows. The inverse Laplace transform of eqn. (1) is performedto estimate the probability density function ƒ_(x) (x) from the measureddata. Next, using eqns. (3)-(5), the average chain length N and thechain length distribution f_(N)(N) are computed. Viscosity η of thehydrocarbon is then estimated using eqn. (5). This traditional analysishas some disadvantages. First, the inverse Laplace transform of themeasured NMR data in eqn. (1) is non-linear due to the non-negativityconstraint on f_(x)(x). See, Venkataramanan et al. 2002, and E. J.Fordham, A. Sezginer, and L. D. Hall, Imaging multiexponentialrelaxation in the (y, log _(e) T ₁) plane, with application to clayfiltration in rock cores, J. Mag. Resonance A, 113:139-150, 1995. Next,it is also mathematically ill-conditioned. Often, regularization orprior information about the expected solution is incorporated into theproblem formulation to make it better conditioned. However, the choiceof the regularization functional as well as the weight given to theprior information is a well-known drawback of this transform due to itsnon-uniqueness. See, W. H. Press, S. A. Teukolsky, and W. T. Vetterling,Numerical Recipes in C, Cambridge University Press, 1992. Thus errors inthe estimation of T₁, T₂, or D distributions are propagated into errorsin the estimation of fluid properties.

SUMMARY

According to some embodiments, a method for estimating a property of amaterial is provided. A measurable change in the material is induced andmeasurements are made of an aspect of the material that tends toexponentially decay following the induction. The induction is repeatedat predetermined intervals, and the material property is estimated basedon measurements of the aspect. The intervals are designed so as toimprove the estimation of the property.

The induction can be a transmitted transverse magnetic pulse as with NMRmeasurements, according to some embodiments. According to someembodiments the induction is the induction of a pressure or temperaturechange in the material. According to some embodiments, the intervalspacing (e.g. in time, in frequency, etc.) is designed according to aformula determined by the property being estimated, and the spacing isnon-uniform in both linear and logarithmic scales. According to someembodiments the interval spacing in time follows a power law and allowsfor the material property to be directly computed from the measurements.

The method can be carried out as part of an oilfield service, and themeasurements can be made downhole. The material can be a fluid, a solidor a mixture. According to some embodiments, the material is one or morerock solids and the material properties being estimated arepetrophysical properties of the rock solids.

According to some embodiments, a system for estimating a property of amaterial is provided that includes an inducer that is adapted to inducea measurable change in the material, and one or more sensors adapted tomeasure an aspect of the material that tends to exponentially decayfollowing an induction. A controller is coupled to the inducer, andadapted and programmed to repeat the induction at predeterminedintervals. A processing system is adapted to estimate the materialproperty based on measurements. The induction intervals are designed soas to improve the estimation of the property.

Further features and advantages will become more readily apparent fromthe following detailed description when taken in conjunction with theaccompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

The present disclosure is further described in the detailed descriptionwhich follows, in reference to the noted plurality of drawings by way ofnon-limiting examples of exemplary embodiments, in which like referencenumerals represent similar parts throughout the several views of thedrawings, and wherein:

FIG. 1 is a plot showing the values of n and μ for different values of ωas determined by equation (8) herein;

FIG. 2 is a plot showing an example of a single exponential acquired atuniform time intervals in the time-domain;

FIG. 3 is a plot showing a non-uniform sampling, according to someembodiments;

FIG. 4 plots examples of non-uniform sampling in time domain that candirectly provide moments of relaxation or diffusion distribution,according to some embodiments;

FIG. 5 is a plot showing the ratio of the error-bar in the momentsobtained from uniform sampling to non-uniform sampling;

FIG. 6A shows an NMR tool being deployed in a wellbore, according toembodiments;

FIG. 6B is a schematic of a processing system used to control, recordand process data from an NMR tool; and

FIG. 7 illustrates another example of a wellsite system in which an NMRtool can be employed, according to some embodiments.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

The following description provides exemplary embodiments only, and isnot intended to limit the scope, applicability, or configuration of thedisclosure. Rather, the following description of the exemplaryembodiments will provide those skilled in the art with an enablingdescription for implementing one or more exemplary embodiments. It beingunderstood that various changes may be made in the function andarrangement of elements without departing from the spirit and scope ofthe invention as set forth in the appended claims.

Specific details are given in the following description to provide athorough understanding of the embodiments. However, it will beunderstood by one of ordinary skill in the art that the embodiments maybe practiced without these specific details. For example, systems,processes, and other elements in the invention may be shown ascomponents in block diagram form in order not to obscure the embodimentsin unnecessary detail. In other instances, well-known processes,structures, and techniques may be shown without unnecessary detail inorder to avoid obscuring the embodiments. Further, like referencenumbers and designations in the various drawings indicated likeelements.

Also, it is noted that individual embodiments may be described as aprocess which is depicted as a flowchart, a flow diagram, a data flowdiagram, a structure diagram, or a block diagram. Although a flowchartmay describe the operations as a sequential process, many of theoperations can be performed in parallel or concurrently. In addition,the order of the operations may be re-arranged. A process may beterminated when its operations are completed, but could have additionalsteps not discussed or included in a figure. Furthermore, not alloperations in any particularly described process may occur in allembodiments. A process may correspond to a method, a function, aprocedure, a subroutine, a subprogram, etc. When a process correspondsto a function, its termination corresponds to a return of the functionto the calling function or the main function.

Furthermore, embodiments of the invention may be implemented, at leastin part, either manually or automatically. Manual or automaticimplementations may be executed, or at least assisted, through the useof machines, hardware, software, firmware, middleware, microcode,hardware description languages, or any combination thereof. Whenimplemented in software, firmware, middleware or microcode, the programcode or code segments to perform the necessary tasks may be stored in amachine readable medium. A processor(s) may perform the necessary tasks.

According to some embodiments, modified pulse sequences are providedthat allow for the direct measurements of moments of relaxation-time ordiffusion distributions. The data from these pulse sequences providesdirect estimates of fluid properties such as average chain length andviscosity of a hydrocarbon. In comparison to the uniformly-spaced pulsesequence, these pulse sequences are faster and have a lower error bar incomputing the fluid properties.

Co-pending U.S. Patent Application No. 61/242,218, filed Sep. 14, 2009(incorporated herein by reference) describes a method to analyze NMRdata that obviates the estimation of the T₁, T₂, or D distributions. Anew mathematical formulation is described where the moments of a randomvariable can be directly computed from the measurement. The method hastwo distinct advantages: the computation of moments is linear and doesnot require prior knowledge or information of the expected distribution.A moment of a random variable x with density function ƒ_(x)(x) isdefined as,

$\begin{matrix}{\left\langle x^{\omega} \right\rangle \equiv \frac{\int_{0}^{\infty}{x^{\omega}{f_{x}(x)}{\mathbb{d}x}}}{\int_{0}^{\infty}{{f_{x}(x)}{\mathbb{d}x}}}} & (6)\end{matrix}$For NMR applications, the variable x corresponds to relaxation times T₁,T₂, or diffusion D. The moments of relaxation times or diffusion can bedirectly computed from the measured magnetization data,

$\begin{matrix}{{\left\langle x^{\omega} \right\rangle = {\frac{\left( {- 1} \right)^{n}}{{\Gamma(\mu)}\phi}{\int_{0}^{\infty}{{t^{\mu - 1}\left\lbrack \frac{\mathbb{d}^{n}{M(t)}}{\mathbb{d}t^{n}} \right\rbrack}{\mathbb{d}t}}}}},{\omega = {\mu - n}}} & (7)\end{matrix}$where Γ( ) represents the Gamma function and

ϕ = M(t = 0) = ∫₀^(∞)f_(x)(x)𝕕x.The contribution of variable ω is in two parts: a real number μ and aninteger n where the mathematical operator t^(μ-1) operates on the n-thderivative of the data. Given a value of ω, the values of n and μ areunique and given as,ω>0:n=0,μ=ωω≦0:ω=μ−n,0<μ≦1,n=[−ω]+1  (8)where [ω] refers to the integral part of real number ω. FIG. 1 is a plot110 showing the values of n and μ for different values of ω asdetermined by above these equations.

According to some embodiments, the computation of moments of relaxationtimes T₁, T₂, or diffusion D, in conjunction with the scaling laws, canlead to greatly improved alternative pulse sequences to specificallytarget or estimate only certain moments of relaxation time/diffusion. Inco-pending U.S. Patent Application Publication No. 2008-0314582(incorporated herein by reference), “targeted” measurements aredescribed as measurements that target a certain specified parameter at aparticular location in the reservoir space. These targeted measurementsare achieved by both hardware and software modifications.

According to some embodiments, the concept of targeted measurements isapplied to NMR data and alternate pulse sequences are described todirectly estimate the moments of relaxation time or diffusioncoefficients. According to some embodiments, alternate pulse sequencesare designed by taking advantage of the specific exponential-kernelstructure in eqn. (1) as well as the computation of moments using eqn.(7). The magnetization data from the new pulse sequences can be directlyinterpreted to obtain fluid properties.

According to some embodiments, further detail will now be provided formodified pulse sequences for measuring NMR data. The non-uniformlyspaced pulse sequences are targeted to specifically measure one of themoments of the relaxation time T₁, T₂, or D distribution. Forsimplicity, consider that the random variable in eqn. (1) is T₁ and aspecific ω-th moment of T₁ is desired, where ω>0. This corresponds tothe right-hand side of the ω axis in FIG. 1. In this case, from eqns.(7) and (8),

$\begin{matrix}{\left\langle T_{1}^{\omega} \right\rangle = {\frac{1}{{\Gamma(\omega)}\phi}{\int_{0}^{\infty}{t^{\omega - 1}{M(t)}{\mathbb{d}t}}}}} & (9)\end{matrix}$We introduce a new variable whereτ=t^(ω)  (10)Therefore, eqn. (9) can be re-written as

$\begin{matrix}{\left\langle T_{1}^{\omega} \right\rangle = {\frac{1}{{\Gamma\left( {\omega + 1} \right)}\phi}{\int_{0}^{\infty}{{M_{\tau}(\tau)}{\mathbb{d}\tau}}}}} & (11)\end{matrix}$where M_(τ)(τ) is the magnetization data in the τ domain. Eqn. (11)indicates that if the data are acquired such that they are uniformlyspaced in the τ domain, then the sum of the measured data directlyprovides the ω-th moment. From eqn. (10), uniform spacing in the τdomain implies non-uniform spacing in the time (t) domain.

We illustrate this with an example. FIG. 2 is a plot showing an exampleof a single exponential acquired at uniform time intervals in thetime-domain. In particular, curve 210 plots a single exponential decayof the form M(t)=e^(−t/T) ¹ in the time-domain as shown in where thedata are uniformly sampled in time. In this example, suppose that the0.5-th moment of T₁ is required. FIG. 3 is a plot showing a non-uniformsampling, according to some embodiments. In particular, using eqn. (10),the data shown by curve 310 are sampled uniformly in the √{square rootover (t)} domain. Curve 310 is an example of a single exponential in thetime-domain into a Guassian in the τ domain with M(t)=e^(−τ) ² ^(/T) ¹ .Using eqn. (11), the sum of measured echoes of the Gaussian directlyprovides

T₁ ^(0.5)

. Thus, if the ω-th moment is desired, then the data need to be samplednon-uniformly in time according to eqn. (10). FIG. 4 plots examples ofnon-uniform sampling in time domain that can directly provide moments ofrelaxation or diffusion distribution, according to some embodiments. Inparticular, plots 410, 412, 414 and 416 represent sampling for ω valuesof 1, 0.75, 0.5, and 0.25 respectively. When 0<Ω<1, this leads to finersampling at the initial part of the data and coarser sampling at thetail end of the data.

The concept of non-uniform sampling is also valid when ω<0. Consider thecase when −1<ω≦0. In this case, eqn. (7) reduces to

$\begin{matrix}{\left\langle T_{1}^{\omega} \right\rangle = {{\frac{- 1}{{\Gamma(\mu)}\phi}{\int_{0}^{\infty}{{t^{\mu - 1}\left\lbrack \frac{\mathbb{d}{M(t)}}{\mathbb{d}t} \right\rbrack}{\mathbb{d}t}\mspace{14mu}{where}\mspace{14mu}\mu}}} = {{1 + \omega} > 0}}} & (12)\end{matrix}$Eqn. (12) can be re-written as

$\begin{matrix}{\left\langle T_{1}^{\omega} \right\rangle = {\frac{- 1}{{\Gamma(\mu)}\phi}{\int_{0}^{\infty}{{t^{\mu - 1}\left\lbrack \frac{\mathbb{d}\left\lbrack {{M(t)} - {M(0)}} \right\rbrack}{\mathbb{d}t} \right\rbrack}{\mathbb{d}t}}}}} & (13)\end{matrix}$Using integration by parts,

$\begin{matrix}\begin{matrix}{\left\langle T_{1}^{\omega} \right\rangle = {{\frac{- 1}{{\Gamma(\mu)}\phi}\left\lbrack {t^{\mu - 1}\left\{ {{M(t)} - {M(0)}} \right\}} \right\rbrack}_{t = 0}^{t = \infty} +}} \\{\frac{\left( {\mu - 1} \right)}{{\Gamma(\mu)}\phi}{\int_{0}^{\propto}{{t^{\mu - 2}\left\lbrack {{M(t)} - {M(0)}} \right\rbrack}{\mathbb{d}t}}}} \\{= {{\frac{M(0)}{\phi}\delta_{\mu\; 1}} + {\frac{\left( {\mu - 1} \right)}{{\Gamma(\mu)}\phi}{\int_{0}^{\infty}{{t^{\mu - 2}\left\lbrack {{M(t)} - {M(0)}} \right\rbrack}{\mathbb{d}t}}}}}}\end{matrix} & \begin{matrix}(14) \\\; \\\; \\\; \\(15)\end{matrix}\end{matrix}$where δ_(μ1)=1 only if μ=1 and is zero otherwise. When μ=1(corresponding to ω=0), the first term in eqn. (15) is 1 (since M(0)=φ)and the second term is zero. This is consistent with the definition ofthe zero-th moment in eqn. (6) with

T₁ ^(ω=0)

=1.

When 0<μ<1, the first term in eqn. (15) is zero. The second term isintegrable at t=0 since [M(t)−M(0)]˜t as t→0. In this case, we introducea new variable τ whereτ=t^(μ=1)  (16)Here, τ→∞ when t→0, and τ→0 when t→∞. Therefore, eqn. (15) becomes

$\begin{matrix}{\left\langle T_{1}^{\omega} \right\rangle = {\frac{- 1}{{\Gamma(\mu)}\phi}{\int_{0}^{\infty}{\left\lbrack {{M_{T}(\tau)} - {M(0)}} \right\rbrack{\mathbb{d}\tau}}}}} & (17)\end{matrix}$Similar to eqn. (11), the area under the integrand [M_(T)−M(0)] can beused to directly estimate the ω-th moment. As shown herein below, theconcept of the pulse sequence for ω>0 and −1<ω≦0 is valid for otherranges of ω as well. For example, when −2<ω≦−1, pulse sequences thatdirectly measure the ω-th moment are obtained by Taylor series expansionof the data with second-order terms in eqn. (13). Higher-order terms inthe Taylor series lead to further negative values of ω. As shown hereinbelow, when ω is not a negative integer,

T₁ ^(ω)

can be obtained by sampling uniformly in the τ domain, whereτ=t^(ω)  (18)When ω is a negative integer, from eqns. (7) and (8), the ω-th moment isobtained by the |ω|-th derivative of the data,

$\begin{matrix}{\left\langle T_{1}^{\omega} \right\rangle = {\frac{\left( {- 1} \right)^{\omega }}{\phi}\frac{\mathbb{d}^{\omega }{M\left( {t = 0} \right)}}{\mathbb{d}t^{\omega }}}} & (19)\end{matrix}$

In addition to computing the moments directly, this modified pulsesequence also enables fluid properties such as the average chain lengthand viscosity to be directly estimated in a simple manner. Re-writingeqns. (3) and (4), the average chain length and viscosity can becomputed from moments of T₁ or T₂ relaxation time or diffusion, as

$\begin{matrix}{\overset{\_}{N} = \left\lbrack {B^{{- 1}/\kappa}\left\langle T_{1}^{1/\kappa} \right\rangle} \right\rbrack^{- \frac{\kappa}{\gamma + \kappa}}} & (20) \\{\overset{\_}{N} = \left\lbrack {A^{{- 1}/v}\left\langle D^{1/v} \right\rangle} \right\rbrack^{- \frac{v}{v + \beta}}} & (21) \\\left. {\eta \propto \left\langle D^{- 1} \right\rangle}\Rightarrow{\eta \propto {\left\lbrack \left\langle T^{\frac{1}{k}} \right\rangle \right\rbrack^{\frac{{\beta\; k} - {\gamma\; v}}{\gamma + k}}\left\langle T_{1}^{- \frac{v}{k}} \right\rangle}} \right. & (22)\end{matrix}$According to eqns. (20) and (21), the average chain length can beestimated from a modified pulse sequence to directly estimate the(1/κ)-th moment of relaxation time or (1/ν)-th moment of diffusion.Similarly, viscosity can be estimated from two modified pulse sequencesto directly estimate the (1/κ)-th and (1/ν)-th moment of relaxationtime. For diffusion data, the kernel in eqn. (1) is e^(−Dt). Using eqn.(10) with ω=1, the negative first moment of diffusion and viscosity areobtained.

In addition to directly providing the parameters of interest, thesepulse sequences are optimized to estimate any desired moments ofrelaxation time and/or diffusion. In comparison to pulse sequences wherethe data is uniformly acquired in time, these pulse sequences can beused to acquire the data faster. For example, given ω>0, the error-barin the ω-th moment of T₁ relaxation is,

$\begin{matrix}{{\sigma_{\langle T_{1}^{\omega}\rangle}❘_{{Uniform}\mspace{14mu}{Sampling}}} = {\frac{\sigma_{\varepsilon}}{{\Gamma(\omega)}\phi}t_{E}^{\omega}\sqrt{\sum\limits_{n = 1}^{N_{e}}n^{{2\omega} - 2}}}} & (23)\end{matrix}$Here N_(e) denotes the total number of echoes obtained from uniformsampling between a given lower and upper limit of relaxation time andt_(E) corresponds to the spacing between the echoes. In the case ofnon-uniform sampling, the error-bar in the ω-th moment is,

$\begin{matrix}{{\sigma_{\langle T_{1}^{\omega}\rangle}❘_{{Non}\text{-}{Uniform}\mspace{14mu}{Sampling}}} = {\frac{\sigma_{\varepsilon}}{{\Gamma\left( {\omega + 1} \right)}\phi}\tau_{E}\sqrt{N_{e}}}} & (24)\end{matrix}$Here τ_(E) corresponds to the spacing between echoes in the τ domain.FIG. 5 is a plot showing the ratio of the error-bar in the momentsobtained from uniform sampling to non-uniform sampling. There are twopoints to note from this curve 510. First, there is a significantadvantage to non-uniform sampling as the error-bar in the computation ofmoments is lower (or at worst equal when ω=1). Second, from eqns. (23)and (24), the standard deviation of noise for uniform sampling has to besmaller than that for non-uniform sampling to obtain a desired error-barin the moments. Since the magnitude of the noise standard deviation inthe data is inversely proportional to the square-root of the acquisitiontime, to achieve the same uncertainty in the parameter the data fornon-uniform sampling can be acquired faster than that for uniformsampling.

Thus, according to some embodiments, a modified pulse sequence isprovided that can directly provide moments of relaxation-time ordiffusion distributions. This pulse sequence can be adapted to thedesired moment of relaxation-time or diffusion coefficient. The datafrom this pulse sequence provides direct estimates of fluid propertiessuch as average chain length and viscosity of a hydrocarbon. In general,in comparison with data acquired uniformly in time, this pulse sequenceis faster and has a lower error bar in computing the fluid properties.

A general expression for the sampling rate of non-uniformly spaced pulsesequence will now be described in further detail, according to someembodiments. When ω is not a negative integer,

T₁ ^(ω)

can be obtained by sampling uniformly in the τ domain, whereτ=t^(ω)  (25)

Proof: For ω>0 and 1<ω<0, the proof has already been demonstrated ineqn. (11) and eqns. (12)-(17) respectively. When −2<ω≦1, from eqn. (8),n=2 and μ=ω+2. In this case, from eqn. (7),

$\begin{matrix}\begin{matrix}{\left\langle T_{1}^{\omega} \right\rangle = {\frac{1}{{\phi\Gamma}(\mu)}{\int_{0}^{\infty}{t^{\mu - 1}\frac{\mathbb{d}^{2}}{\mathbb{d}t^{2}}{M(t)}{\mathbb{d}t}}}}} \\{= {\frac{1}{{\phi\Gamma}(\mu)}{\int_{0}^{\infty}{t^{\mu - 1}{\frac{\mathbb{d}^{2}}{\mathbb{d}t^{2}}\left\lbrack {{M(t)} - {M(0)} - {{tM}^{(1)}(0)}} \right\rbrack}{\mathbb{d}t}}}}} \\{= {{\frac{1}{{\phi\Gamma}(\mu)}\left\lbrack {t^{\mu - 1}\frac{\mathbb{d}}{\mathbb{d}t}\left\{ {{M(t)} - {M(0)} - {{tM}^{(1)}(0)}} \right\}} \right\rbrack}_{t = 0}^{t = \infty} -}} \\{\frac{\left( {\mu - 1} \right)}{{\phi\Gamma}(\mu)}{\int_{0}^{\infty}{t^{\mu - 2}{\frac{\mathbb{d}}{\mathbb{d}t}\left\lbrack {{M(t)} - {M(0)} - {{tM}^{(1)}(0)}} \right\rbrack}{\mathbb{d}t}}}} \\{= {{{- \frac{1}{\phi}}{M^{(1)}(0)}\delta_{\mu\; 1}} - \frac{1}{{\phi\Gamma}\left( {\mu - 1} \right)}}} \\{\left\lbrack {t^{\mu - 2}\left\{ {{M(t)} - {M(0)} - {{tM}^{(1)}(0)}} \right\}} \right\rbrack_{t = 0}^{t = \infty} +} \\{\frac{\left( {\mu - 2} \right)}{{\phi\Gamma}\left( {\mu - 1} \right)}{\int_{0}^{\infty}{{t^{\mu - 3}\left\lbrack {{M(t)} - {M(0)} - {{tM}^{(1)}(0)}} \right\rbrack}{\mathbb{d}t}}}} \\{= {{{- \frac{1}{\phi}}{M^{(1)}(0)}\delta_{\mu\; 1}} + {\frac{\left( {\mu - 2} \right)}{{\phi\Gamma}\left( {\mu - 1} \right)}{\int_{0}^{\infty}t^{\mu - 3}}}}} \\{\left\lbrack {{M(t)} - {M(0)} - {{tM}^{(1)}(0)}} \right\rbrack{\mathbb{d}t}} \\{= {{{- \frac{1}{\phi}}{M^{(1)}(0)}\delta_{\mu\; 1}} + {\frac{1}{{\phi\Gamma}(\omega)}{\int_{0}^{\infty}t^{\omega - 1}}}}} \\{\left\lbrack {{M(t)} - {M(0)} - {{tM}^{(1)}(0)}} \right\rbrack{\mathbb{d}t}}\end{matrix} & (26)\end{matrix}$where δ_(μ1)=1 only when μ=1 and is zero otherwise and

${M^{1}(t)} = {\frac{\mathbb{d}{M(t)}}{\mathbb{d}t}.}$When μ=1 (corresponding to ω=1), the first term in eqn. (26) correspondsto the first derivative of the data at t=0 and is consistent with thedefinition of the negative first moment. In this case, the second termis zero. When −2<μ<1, the first term in eqn. (26) is zero. The secondterm is integrable at t=0. Using eqn. (25), the area under the integrand[M(t)−M(0)−tM⁽¹⁾(0)] can be used to directly estimate the ω-th moment.

In general, when n≧1, eqn. (7) can be written as

$\begin{matrix}\begin{matrix}{\left\langle T_{1}^{\omega} \right\rangle = {\frac{\left( {- 1} \right)^{n}}{{\phi\Gamma}(\mu)}{\int_{0}^{\infty}{t^{\mu - 1}\frac{\mathbb{d}^{n}}{\mathbb{d}t^{n}}{M(t)}{\mathbb{d}t}}}}} \\{= {\frac{\left( {- 1} \right)^{n}}{{\phi\Gamma}(\mu)}{\int_{0}^{\infty}{t^{\mu - 1}{M^{(n)}(t)}{\mathbb{d}t}}}}} \\{= {{\frac{\left( {- 1} \right)^{n - 1}}{\phi}{M^{({n - 1})}(0)}\delta_{\mu\; 1}} + {\frac{1}{{\phi\Gamma}\left( {\mu - n} \right)}{\int_{0}^{\infty}t^{\mu - n - 1}}}}} \\{\left\lbrack {{M(t)} - {\sum\limits_{m = 0}^{n - 1}{\frac{t^{m}}{m!}{M^{(m)}(0)}}}} \right\rbrack{\mathbb{d}t}} \\{= {{\frac{\left( {- 1} \right)^{n - 1}}{\phi}{M^{({n - 1})}(0)}\delta_{\mu\; 1}} + {\frac{1}{{\phi\Gamma}(\omega)}{\int_{0}^{\infty}t^{\omega - 1}}}}} \\{\left\lbrack {{M(t)} - {\sum\limits_{m = 0}^{n - 1}{\frac{t^{m}}{m!}{M^{(m)}(0)}}}} \right\rbrack{\mathbb{d}t}}\end{matrix} & (27) \\{where} & \; \\{{{M^{(m)}(t)} = {\frac{\mathbb{d}^{m}}{\mathbb{d}t^{m}}{M(t)}}}{and}} & (28) \\{{M^{(0)}(t)} = {M(t)}} & (29)\end{matrix}$Note that:

$\begin{matrix}{{M(t)} - {\sum\limits_{m = 0}^{n - 1}{\frac{t^{m}}{m!}\left. {M^{(m)}(0)}\longrightarrow t^{n} \right.\mspace{25mu}{as}\mspace{25mu}\left. t\longrightarrow 0 \right.}}} & (30)\end{matrix}$

As before, the first term in eqn. (27) is valid only for negativeinteger values of ω and corresponds to the |ω|-th derivative of thedata. For negative values of ω where n<ω<(n−1), the first term in eqn.(27) is zero. Using eqn. (25), the area under the integrand

$\left\lbrack {{M(t)} - {\sum\limits_{m = 0}^{n - 1}{\frac{t^{m}}{m!}{M^{(m)}(0)}}}} \right\rbrack$in the τ domain directly provides the ω-th moment of relaxation time.

FIG. 6A shows an NMR tool being deployed in a wellbore, according toembodiments. Wireline truck 610 is deploying wireline cable 612 intowell 630 via well head 620. Wireline tool 640 is disposed on the end ofthe cable 612 in a subterranean formation 600. Tool 640 includes an NMRtool, such as Schlumberger's Combinable Magnetic Resonance (CMR) Tool,or Schlumberger's MR Scanner Tool. According to some embodiments the NMRtool is combined with other downhole tools such as Schlumberger'sModular Formation Dynamics Tester downhole sampling tool. According tosome embodiments, the NMR tool uses a pulse sequence as describedherein. Data from the NMR tool can be recorded and/or processed downholewithin tool 640, and/or can be transmitted to wireline truck 610 forrecording and/or processing. According to embodiments, the NMR tool canbe controlled, including the selection of pulse sequences, locally withprocessing systems within tool 640 and/or from the surface usingprocessing systems for example in wireline truck 610.

FIG. 6B is a schematic of a processing system used to control, recordand process data from an NMR tool. Processing system 650 includes one ormore central processing units 674, storage system 672, communicationsand input/output modules 670, a user display 676 and a user input system678. Input/output modules 670 include modules to communicate with andcontrol the NMR tool such that modified pulse sequences can be used asdescribed herein.

FIG. 7 illustrates another example of a wellsite system in which an NMRtool can be employed, according to some embodiments. The wellsite can beonshore or offshore. In this exemplary system, a borehole 711 is formedin subsurface formations by rotary drilling in a manner that is wellknown. Embodiments of the invention can also use directional drilling,as will be described hereinafter.

A drill string 712 is suspended within the borehole 711 and has a bottomhole assembly 700 which includes a drill bit 705 at its lower end. Thesurface system includes platform and derrick assembly 710 positionedover the borehole 711, the assembly 710 including a rotary table 716,kelly 717, hook 718 and rotary swivel 719. The drill string 712 isrotated by the rotary table 716, energized by means not shown, whichengages the kelly 717 at the upper end of the drill string. The drillstring 712 is suspended from a hook 718, attached to a traveling block(also not shown), through the kelly 717 and a rotary swivel 719 whichpermits rotation of the drill string relative to the hook. As is wellknown, a top drive system could alternatively be used.

In the example of this embodiment, the surface system further includesdrilling fluid or mud 726 stored in a pit 727 formed at the well site. Apump 729 delivers the drilling fluid 726 to the interior of the drillstring 712 via a port in the swivel 719, causing the drilling fluid toflow downwardly through the drill string 712 as indicated by thedirectional arrow 708. The drilling fluid exits the drill string 712 viaports in the drill bit 705, and then circulates upwardly through theannulus region between the outside of the drill string and the wall ofthe borehole, as indicated by the directional arrows 709. In this wellknown manner, the drilling fluid lubricates the drill bit 705 andcarries formation cuttings up to the surface as it is returned to thepit 727 for recirculation.

The bottom hole assembly 700 of the illustrated embodiment alogging-while-drilling (LWD) module 720, a measuring-while-drilling(MWD) module 730, a roto-steerable system and motor, and drill bit 705.

The LWD module 720 is housed in a special type of drill collar, as isknown in the art, and can contain one or a plurality of known types oflogging tools. It will also be understood that more than one LWD and/orMWD module can be employed, e.g. as represented at 720A. (References,throughout, to a module at the position of 720 can alternatively mean amodule at the position of 720A as well.) The LWD module includescapabilities for measuring, processing, and storing information, as wellas for communicating with the surface equipment. In the presentembodiment, the LWD module includes an NMR tool such as Schlumberger'sProVISION NMR tool.

The MWD module 730 is also housed in a special type of drill collar, asis known in the art, and can contain one or more devices for measuringcharacteristics of the drill string and drill bit. The MWD tool furtherincludes an apparatus (not shown) for generating electrical power to thedownhole system. This may typically include a mud turbine generatorpowered by the flow of the drilling fluid, it being understood thatother power and/or battery systems may be employed. In the presentembodiment, the MWD module includes one or more of the following typesof measuring devices: a weight-on-bit measuring device, a torquemeasuring device, a vibration measuring device, a shock measuringdevice, a stick slip measuring device, a direction measuring device, andan inclination measuring device.

According to some embodiments, the techniques described herein areapplied to estimate properties from other types of excitations than NMR.For example the techniques can be applied to other types of data whereexcitations induce an exponential decay. According to some embodiments,the techniques are applied to inducing a non-uniform sequence ofpressure changes in a material. According to some other embodiments, thetechniques are applied to induce a non-uniform sequence of temperaturechanges in a material.

While many examples have been described herein with respect to designinginterval spacings in the time domain, according to some embodiments, thetechniques described herein are applied to interval spacings in otherdomains, such as designing interval spacings in the frequency domain.

Whereas many alterations and modifications of the present disclosurewill no doubt become apparent to a person of ordinary skill in the artafter having read the foregoing description, it is to be understood thatthe particular embodiments shown and described by way of illustrationare in no way intended to be considered limiting. Further, thedisclosure has been described with reference to particular preferredembodiments, but variations within the spirit and scope of thedisclosure will occur to those skilled in the art. It is noted that theforegoing examples have been provided merely for the purpose ofexplanation and are in no way to be construed as limiting of the presentdisclosure. While the present disclosure has been described withreference to exemplary embodiments, it is understood that the words,which have been used herein, are words of description and illustration,rather than words of limitation. Changes may be made, within the purviewof the appended claims, as presently stated and as amended, withoutdeparting from the scope and spirit of the present disclosure in itsaspects. Although the present disclosure has been described herein withreference to particular means, materials and embodiments, the presentdisclosure is not intended to be limited to the particulars disclosedherein; rather, the present disclosure extends to all functionallyequivalent structures, methods and uses, such as are within the scope ofthe appended claims.

What is claimed is:
 1. A method of estimating a property of a materialcomprising: inducing a measurable change in the material; makingmeasurements of a material response to the induction, the materialresponse exponentially decaying following the induction; repeating theinduction at predetermined intervals; estimating the property of thematerial at least in part on the measurements of the exponentiallydecaying material response, the predetermined intervals being designedand modified based on the measurement of the exponentially decayingmaterial response so as to improve the estimation of the property.
 2. Amethod according to claim 1 wherein the induction is an excitation ofthe material.
 3. A method according to claim 2 wherein the induction isa transmitted transverse magnetic pulse.
 4. A method according to claim2 wherein the induction is an applied pressure pulse.
 5. A methodaccording to claim 2 wherein the induction is the induction of atemperature change in the material.
 6. A method according to claim 1wherein an interval spacing in time is designed according to a formuladetermined by the property being estimated.
 7. A method according toclaim 1 wherein an interval spacing in a frequency domain is designedaccording to a formula determined by the property being estimated.
 8. Amethod according to claim 1 wherein an interval spacing of the inductionis non-uniform in both linear and logarithmic scales.
 9. A methodaccording to claim 1 wherein an interval spacing of the induction intime is predetermined using a power law.
 10. A method according to claim1 wherein an interval spacing of the induction in a frequency domain ispredetermined using a power law.
 11. A method according to claim 1wherein an interval spacing in time is selected for the property to bedirectly computed from the measurements.
 12. A method according to claim1 wherein an interval spacing in a frequency domain is selected for theproperty to be directly computed from the measurements.
 13. A methodaccording to claim 1 wherein the method is carried out as part of one ormore oilfield services.
 14. A method according to claim 13 wherein themeasurements are made downhole.
 15. A method according to claim 13wherein the measurements are made on the surface.
 16. A method accordingto claim 1 wherein the material is a fluid.
 17. A method according toclaim 16 wherein the properties are fluid properties.
 18. A methodaccording to claim 1 wherein the material is one or more rock solids andthe properties are petrophysical properties of the one or more rocksolids.
 19. A method according to claim 18 wherein the induction isnuclear magnetic resonance and the property is a moment ofrelaxation-time or diffusion distribution.
 20. A method according toclaim 1 wherein the material is a mixture of solid and liquid.
 21. Amethod according to claim 20 wherein the properties are fluid andpetrophysical properties in porous solids.
 22. A system for estimating aproperty of a material comprising: an inducer that is adapted to inducea measurable change in the material; one or more sensors adapted tomeasure a material response that tends to exponentially decay followingthe induction; a controller coupled to the inducer, adapted andprogrammed to repeat the induction at predetermined intervals; and aprocessing system adapted to estimate the property of the material atleast in part on the one or more sensor measurements, wherein thepredetermined intervals are designed and modified based on themeasurement of the exponentially decaying material response so as toimprove the estimation of the property.
 23. A system according to claim22 wherein the inducer induces an excitation of the material.
 24. Asystem according to claim 23 wherein the inducer transmits a transversemagnetic pulse using an antenna.
 25. A system according to claim 23wherein the inducer applies a pressure pulse.
 26. A system according toclaim 23 wherein the inducer induces a temperature change in thematerial.
 27. A system according to claim 22 wherein an interval spacingin time is designed according to a formula determined by the property tobe estimated.
 28. A system according to claim 22 wherein an intervalspacing in a frequency domain is designed according to a formuladetermined by the property to be estimated.
 29. A system according toclaim 22 wherein the intervals are non-uniform in both linear andlogarithmic scales.
 30. A system according to claim 22 wherein theintervals is predetermined using a power law.
 31. A system according toclaim 22 wherein the intervals are selected for the property to bedirectly computed from the measurements.
 32. A system according to claim22 wherein the system is carried out as part of one or more oilfieldservices.
 33. A system according to claim 32 further comprising adownhole deployable tool body that houses the inducer and the sensors.34. A system according to claim 32 further comprising a surfaceinstrument that houses the inducer and the sensors.
 35. A systemaccording to claim 22 wherein the material is a fluid.
 36. A systemaccording to claim 35 wherein the properties are fluid properties.
 37. Asystem according to claim 22 wherein the material is one or more rocksolids and the properties are petrophysical properties of the one ormore rock solids.
 38. A system according to claim 37 wherein theinduction is nuclear magnetic resonance and the property is a moment ofrelaxation-time or diffusion distribution.
 39. A system according toclaim 22 wherein the material is a mixture of solid and liquid.
 40. Amethod according to claim 39 wherein the properties are fluid propertiesin porous solids.